Extracting Square Roots © 2002 by Shawna Haider. Solving Quadratic Equations by Factoring Let's solve the equation First you need to get it in what we

Extracting Square Roots

© 2002 by Shawna Haider

Solving Quadratic Equations by Factoring

Let's solve the equation 1872 xx

First you need to get it in what we call "quadratic form" which means 02 cbxax

need this to be 0

1

ok

-18 -18

01872 xxSo we have Now let's factor the left hand side 029 xx

Now set each factor = 0 and solve for each answer.02or 09 xx

2or 9 xx

02 cbxaxok

252 t

Extracting Square Roots

The idea behind this method is when you have some "stuff" squared that you can get by itself on the left hand side of the equation (no other variables on the right hand side), you can then take the square root of each side to cancel out the square.

1255 2 tGet the "squared stuff" alone which in this case is the t 2

5 5

25

Now square root each side. Since you loose any negative sign when you square something, both the + and – of the number would solve the equation so you must do both.

252 t

5t

4

492 u

Let's try another one

494 2 uGet the "squared stuff" alone which in this case is the u 2

4 4Now square root each side and

DON'T FORGET BOTH THE + AND –4

492 u

Remember with a fraction you can square root the top and square root the bottom

4

492 u

2

7u

5012 2 x

Another Example

5012 2 xGet the "squared stuff" alone which in this case is the stuff in the parenthesis and it is alone.

Now square root each side and

DON'T FORGET BOTH THE + AND –

Let's simplify the radical2512 x

25 · 2

Now solve for x-1 -1

2512 x2 2 2

251x

2532 2 y

One Last Example

02532 2 yGet the "squared stuff" alone which in this case is the stuff in the parenthesis.

Now square root each side and

DON'T FORGET BOTH THE + AND –

This will give you an iiy 532 Now solve for y+3 +3

iy 532 2 2 2

53 iy

-25 -25

Method 4: The Quadratic Formula

The Quadratic Formula is a formula that can solve any quadratic, but it is best used for equations that cannot be factored or when completing the square requires the use of fractions. It is the most complicated method of the four methods.

Do you want to see where the formula comes from?

The Quadratic Formula2 0ax bx c

2 0ax bx c

a a a a

2 b cx x

a a

2

24

b

a

2

24

b

a

2 2

2

4

2 4

b b acx

a a

2 4

2 2

b b acx

a a

2 4

2 2

b b acx

a a

2 4

2

b b acx

a

This formula comes from completing the square of a quadratic written in standard form

1. Divide by a

2. Subtract c/a and add half of b/a squared

3. Factor left side, combine right side

4. Square root each side

5. Simplify radical

6. Get x alone

7. Simplify right hand side

The Quadratic Formula24 2 5 0x x

2 4 80

8x

Solve the equation

1. Identify a, b, c

2. Plug into the formula

5. Simplify

6. Simplify

7. Simplify radical

2 4

2

b b acx

a

a = 4b= -2c = 5

(4)(2)2 2

(4)

(5)

2 76

8x

2 2 19

8

ix

Notice the solutions are complex!

8. Simplify final answer, if possible

2 2 19

8

ix

1 19

4

ix

=4•19

Meaning:

0 x-intercepts,

2 complex solutions

The Quadratic Formula2 1 0x x Solve the equation

1. Identify a, b, c

2. Plug into the formula

5. Simplify

6. Simplify

7. Simplify radical

2 4

2

b b acx

a

a =b= c =

( )( )2

( )

( )

8. Simplify final answer, if possible

Another example

Solve the equation 22 4 5 0x x

Completing the Square

• Isolate the terms with variables on one side of the equation, and arrange them in descending order.

• Divide both sides by the coefficient of x² if that coefficient is not 1.

• Complete the square by taking half of the coefficient of x and adding its square to both sides.

• Express the trinomial as the square of a binomial (factor the trinomial) and simplify the other side.

• Use the principle of square roots (find the square roots of both sides).

• Solve for x by adding or subtracting on both sides.

Example Only variable terms are on the left side. Subtracting 4 to both sides. We can now complete the square on the left side.

2 6 4 0x x

2 6 ____ 4x x Completing the square:½(6) = 3 and (3)² = 9

2 6 9 4 9x x Factoring and simplifying

2( 3) 5x Using the principle of square roots.

2( 3) 5x 3 5x

Completing the Square: Use #1

This method is used for quadratics that do not factor, although it can be used to solve any kind of quadratic function.

1. Get the x2 and x term on one side and the constant term on the other side of the equation.

2. To “complete the square,”, add “half of b squared” to each side. You will make a perfect square trinomial when you do this.

3 14x

2 6 5 0x x

2 6 5x x

3. Factor the trinomial

9 92 6 9 14x x

2( 3) 14x

3 14x 4. Apply the square root and solve for x

Completing the Square: Use #2

By completing the square, we can take any equation in standard form and find its equation in vertex form: y = a(x-h)2 + k

1. Get the x2 and x term on one side and the constant term on the other side of the equation.

2. To “complete the square,”, add “half of b squared” to each side. You will make a perfect square trinomial when you do this.

( 3, 14)

2 6 5 0x x

2 6 5x x

3. Factor the trinomial.

9 92 6 9 14x x

2( 3) 14x 4. Write in standard form.

2( 3) 14y x

What is the vertex?

Which method should you use?Solve 2 6 5 0x x

a. (x+1)(x+5) b. x = -1, x = -5 c. x = 1, x = 5 d. no sol

Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem?

( )( ) 0x x

Which method should you use?Solve 2 6 5 0x x

a. (x+1)(x+5) b. x = -1, x = -5 c. x = 1, x = 5 d. no sol


( )( ) 0x x

Which method should you use?

Solve 22 1x

a. b.

c. no sol because you cannot factor it

1

2x 2

2x


( )( ) 0x x


Solve 24 2 1 0x x

a. b.

c. d.

2 8

8

ix

2 8

8x

2 2 2

8

ix

1 2

4

ix


( )( ) 0x x



29 81 0x a. b. c. 3x 9x 81x

( )( ) 0x x

Applications of Quadratic Equations

© 2002 by Shawna Haider

P = $500 and A = $572.45

COMPOUND INTEREST

Let's substitute the values we are given for P and A

Solve this equation for r

21 rPA Amount in account after two years

Principal Amount you deposit

Interest rate as a decimal

500500 2150045.572 r

211449.1 r

Square root both sides but don't need negative because interest rate won't be negative

r 107.1 %707.107.1 r

PYTHAGOREAN THEOREM

An L-shaped sidewalk from building A to building B on a college campus is 200 feet long. By cutting diagonally across the grass, students shorten the walking distance to 150 feet. What are the lengths of the two legs of the sidewalk? Draw a picture:

x

200-x

If first part of sidewalk is x and total is 200 then second part is 200 - x A

B

150

222 cba

Using the theorem: 222 150200 xx Multiply out

2250040040000 22 xxxcontinued on next slide

1

2250040040000 22 xxx get everything on one side = 0

0175004002 2 xx divide all terms by 2

087502002 xx use the quadratic formula to solve

12

875014200200 2 x

225100 x 6.64or 4.135 xx

200 - 134.5 = 64.6 so doesn't matter which you choose, the two lengths are 135.4 meters and 64.6 meters.

After how many seconds will the height be 11 feet?

Height of a tennis ball

A tennis ball is tossed vertically upward from a height of 5 feet according to the height equation

where h is the height of the tennis ball in feet and t is the time in seconds.

,52116 2 tth

,5211611 2 ttGet everything on one side = 0 and factor or quadratic formula.

062116 2 tt-11 -11

162

61642121 2

t32

5721

So there are two answers:(use a calculator to find them

making sure to put parenthesis around the numerator)

t = .42 seconds or .89 seconds.

When will the tennis ball hit the ground?

,52116 2 tth

What will the height be when it is on the ground? h = 0

,521160 2 tt

162

51642121 2

t32

76121

So there are two answers: (use a calculator to find them)

t = - 0.21 or 1.52 seconds (throw out the negative one)

Average Speed

Let's make a table with the information

first part

second part

distance rate time

100

135

r t

r - 5 5 - t

If you used t hours for the first part of the trip, then the total 5 minus the t would be the time left for the second part.

A truck traveled the first 100 miles of a trip at one speed and the last 135 miles at an average speed of 5 miles per hour less. If the entire trip took 5 hours, what was the average speed for the first part of the trip?

first part

second part

distance rate time

100

135

r t

r - 5 5 - t

Distance = rate x timeUse this formula to get an equation for each part of trip

100 = r t 135 = (r - 5)(5 - t)Solve first equation for t and substitute in second equation

r r

r

100

rr

10055135

rr

10055135

FOIL the right hand side

rr

500251005135 Multiply all terms by r

to get rid of fractionsr r r r

r

05002605 2 rr Combine like terms and get everything on one side

Divide everything by 5 0100522 rr

Factor or quadratic formula

0250 rr

So r = 50 mph since r = 2 wouldn't work for second part where rate is r –5 and that would be –3 if r was 2.

2or 50 rr

Documents

Extracting Square Roots © 2002 by Shawna Haider. Solving Quadratic Equations by Factoring Let's solve the equation First you need to get it in what we